\section{Reduction to basic algebras and graded reconstruction}
\label{sec:graded-category}

We begin by reducing the existence of a normed multiplicative basis to the
case of basic algebras. We then pass from the category of indecomposable
modules to its radical grading and reconstruct the standard form $A_0$ from
the graded projective objects. Both constructions, together with the
normalization of $A_0$ in the next two sections, apply in arbitrary
characteristic.

\subsection{Multiplicative bases and matrix units}
\label{subsec:nonbasic-basis}

The following lemma allows us to assume throughout the construction that
$A$ is basic.

\begin{lemma}
\label{lem:nonbasic-basis}
Let $A$ be a finite-dimensional algebra over an arbitrary field, and let
$B^{\mathrm{bas}}$ be its basic algebra. If $B^{\mathrm{bas}}$ has a normed multiplicative basis, then
so does $A$.
\end{lemma}
\begin{proof}
Let $\{e_i\}$ be the primitive orthogonal idempotents in the given normed
multiplicative basis of $B^{\mathrm{bas}}$. Each basis element $b$ belongs to
a unique corner $e_iB^{\mathrm{bas}}e_j$, since its decomposition
$b=\sum_{i,j}e_i b e_j$ expresses it as a sum of basis elements in distinct
corners. By Morita equivalence, we may identify $A$ with a block matrix
algebra whose $(i,j)$-block is
$\operatorname{Mat}_{n_i\times n_j}(e_iB^{\mathrm{bas}}e_j)$ for suitable
positive integers $n_i$. Taking the matrices with a single nonzero entry
drawn from the given basis of $B^{\mathrm{bas}}$, we obtain a basis of $A$
that is multiplicative by the matrix-unit multiplication rule, and whose
diagonal copies of the $e_i$ form a complete set of primitive orthogonal
idempotents. Because the $(i,j)$-block of $(\rad A)^n$ is
$\operatorname{Mat}_{n_i\times n_j}(e_i(\rad B^{\mathrm{bas}})^n e_j)$, this
basis is adapted to every radical power and is therefore normed.
\end{proof}

\subsection{Radical layers and mesh relations}

We filter the morphisms in $\ind(\modu A)$ by powers of the categorical
radical, that is, by sums of composites of nonisomorphisms between
indecomposable modules. Because intermediate modules in such a composite need
not be projective, the induced filtration satisfies
$(\rad A)^n(x,y)\subseteq\JJ^n(P_y,P_x)=F^nA(x,y)$, and this inclusion
can be strict; Example~\ref{ex:dual-number-filtration} computes the difference
for the dual numbers below.

Let $A$ be a basic representation-finite algebra over $k$, and let $\JJ$
denote the categorical Jacobson radical of $\ind(\modu A)$. Since $A$ is
representation-finite, the direct sum of a representative set of
indecomposable modules is a finite-dimensional additive generator; its
endomorphism algebra is finite-dimensional, and $\JJ$ consists of the matrix
entries of its Jacobson radical, so $\JJ$ is nilpotent. We define the
\emph{radical grading} of $\ind(\modu A)$ to be the graded category $\C$
with the same objects as $\ind(\modu A)$ and morphism spaces
\[
 \C(X,Y)=\bigoplus_{n\geq0}\JJ^n(X,Y)/\JJ^{n+1}(X,Y),
\]
writing $\C_n(X,Y)=\JJ^n(X,Y)/\JJ^{n+1}(X,Y)$ for the homogeneous component of
degree $n$. For a morphism $f\in \JJ^n(X,Y)\setminus \JJ^{n+1}(X,Y)$, its
\emph{initial form} is its residue class in $\C_n(X,Y)$. Composition in $\C$
is induced by composition in $\ind(\modu A)$. Since the endomorphism ring of
each indecomposable module is local with residue field $k$, we have
$\C_0(X,X)=k\cdot 1_X$ and $\C_0(X,Y)=0$ for $X\ne Y$.

\begin{example}
\label{ex:dual-number-filtration}
Let $A=k[t]/(t^2)$ be the algebra of dual numbers. Up to isomorphism, its
indecomposable modules are $P=A$ and $S=A/(t)$, connected by the canonical
projection $\pi:P\twoheadrightarrow S$ and the inclusion $\iota:S\hookrightarrow P$
sending $1+(t)$ to $t$. In path order, the multiplication-by-$t$
endomorphism of $P$ factors as
\[
 P\xrightarrow{\pi}S\xrightarrow{\iota}P,
 \qquad \pi\iota=t,\qquad \iota\pi=0.
\]
Consequently, $\JJ$ is spanned by $\pi$, $\iota$, and $t$, while $\JJ^2$ is
spanned by $t$ and $\JJ^3=0$. On the projective endomorphism ring $A=\End_A(P)^{\op}$,
the categorical radical filtration is therefore
\[
 A\supset \JJ(P,P)=kt=\JJ^2(P,P)\supset \JJ^3(P,P)=0,
\]
whereas $(\rad A)^2=0$. The associated graded algebra of this filtered endomorphism ring is again
isomorphic to $k[t]/(t^2)$, but its radical generator $t$ sits in
categorical degree two, reflecting its factorization through the
nonprojective simple module $S$.
\end{example}

Let us recall how the Auslander--Reiten quiver $\Gamma_A$ and its mesh
relations arise in this setting. The vertices of $\Gamma_A$ are the objects
of $\ind(\modu A)$, and the arrows $X\to Y$ index a chosen basis of the
space $\Irr(X,Y)=\JJ(X,Y)/\JJ^2(X,Y)$ of irreducible morphisms. For each
nonprojective indecomposable module $Y$, choose an almost split sequence
\[
 0\longrightarrow\tau Y\longrightarrow E_Y
   \longrightarrow Y\longrightarrow0.
\]
If $X$ occurs with multiplicity $m_X$ in $E_Y$, the two maps of the
sequence have $m_X$ components $\tau Y\to X$ and $X\to Y$, respectively.
Modulo $\JJ^2$, these form bases of $\Irr(\tau Y,X)$ and $\Irr(X,Y)$. The next
well-known lemma ensures that each multiplicity is at most one, so that these
components differ from the chosen arrows only by nonzero scalars.

\begin{lemma}
\label{lem:square-free}
Let $A$ be a basic representation-finite algebra over an algebraically
closed field. Its Auslander--Reiten quiver $\Gamma_A$ has no loops and at
most one arrow between each ordered pair of vertices.
\end{lemma}
\begin{proof}
Factoring an irreducible morphism between indecomposable modules through its
image shows that it is either a proper monomorphism or a proper epimorphism;
its source and target therefore have different dimensions, which excludes
loops. If there were two arrows $X\to Y$ with $\dim X>\dim Y$, then $Y$ would
be nonprojective and the middle term of the almost split sequence ending at
$Y$ would contain $X^{\oplus2}$ as a direct summand. Consequently,
\[
 \dim\tau Y\geq2\dim X-\dim Y>\dim X,
\]
and there would again be two arrows $\tau Y\to X$. Iterating this step
produces indecomposable modules of unbounded dimension, contradicting finite
representation type; the case $\dim X<\dim Y$ is dual.
\end{proof}

Thanks to Lemma~\ref{lem:square-free}, we can formulate the mesh category of
$\Gamma_A$ using scalar coefficients rather than matrices.

\begin{definition}
\label{def:translation-quiver}
A \emph{translation quiver} is a quiver $Q$ equipped with a bijection $\tau$
from a subset of vertices, called \emph{nonprojective}, onto a subset of
vertices, called \emph{noninjective}, together with a bijection $\sigma$
from the set of arrows ending at nonprojective vertices onto the set of
arrows starting at noninjective vertices, such that $\sigma a:\tau x\to y$
for every arrow $a:y\to x$ with $x$ nonprojective. We call $\sigma a$ the
\emph{mate} of $a$. Vertices outside the domain (respectively, image) of
$\tau$ are called \emph{projective} (respectively, \emph{injective}). When
both $x$ and $y$ are nonprojective, we write
$\tau a=\sigma^2a:\tau y\to\tau x$. The translation quiver $Q$ is
\emph{stable} if every vertex is both nonprojective and noninjective.
\end{definition}

\begin{definition}
\label{def:scalar-mesh-category}
Let $Q$ be a translation quiver with at most one arrow between each ordered
pair of vertices. Given a family $c=(c_{x,a})$ of nonzero scalars
$c_{x,a}\in k^*$ indexed by nonprojective vertices $x$ and arrows $a:y\to x$
ending at $x$, the associated \emph{scalar mesh category} is the path-length
graded quotient category
\[
 C(Q,c)=kQ/\langle r_x:x\text{ nonprojective}\rangle,
 \qquad r_x=\sum_{a:y\to x}c_{x,a}(\sigma a)a,
\]
where $kQ$ denotes the $k$-linear path category of $Q$.
\end{definition}

Multiplying a mesh relation $r_x$ by a nonzero scalar leaves the relation
ideal unchanged, while rescaling the arrows of $Q$ rescales the coefficients
$c_{x,a}$ and induces a graded isomorphism between the corresponding scalar
mesh categories.

The translation on $\Gamma_A$ is the Auslander--Reiten translate, and
the mate $\sigma a:\tau Y\to X$ of $a:X\to Y$ corresponds to the
same summand $X$ in the almost split sequence ending at $Y$.
For $Q=\Gamma_A$, choose a basis element in each one-dimensional space
$\Irr(X,Y)$ to serve as the unique arrow $a:X\to Y$. For each nonprojective
indecomposable module $Y$, the components of the first and second maps of
the chosen almost split sequence ending at $Y$ take the form
$u_a(\sigma a)$ and $v_a a$ modulo $\JJ^2$, with $u_a,v_a\in k^*$. Since the
composite of the two maps in the almost split sequence is zero, passing to
degree two in $\C$ yields the mesh relation
\begin{equation}
 r_Y=\sum_{a:X\to Y}c_{Y,a}(\sigma a)a,
 \qquad c_{Y,a}=u_av_a.
 \label{eq:scalar-mesh}
\end{equation}
We write $k(\Gamma_A)$ for the scalar mesh category $C(\Gamma_A,c)$ defined
by these coefficients.

Up to isomorphism fixing the objects, $k(\Gamma_A)$ is independent of both
the chosen arrow representatives and the chosen almost split sequences.
Indeed, any isomorphism between two almost split sequences ending at $Y$
induces nonzero scalars $\xi,\eta\in k^*$ on the end terms $\tau Y$ and
$Y$ modulo the radicals of their endomorphism rings. Because the indecomposable summands of the middle term are pairwise
nonisomorphic by Lemma~\ref{lem:square-free}, the automorphism of the middle
term acts by scalars $\lambda_a\in k^*$ on those summands modulo radical
maps. Composing these radical maps with either map of the almost split
sequence gives elements of $\JJ^2$.
The coefficients therefore transform according to
\[
 u'_a=\lambda_a u_a\xi^{-1},\qquad
 v'_a=\eta v_a\lambda_a^{-1},\qquad
 c'_{Y,a}=\eta\xi^{-1}c_{Y,a},
\]
which simply rescales the entire relation $r_Y$ by $\eta\xi^{-1}$. Moreover,
Lemma~\ref{lem:unit-mesh-coefficients} will show that the arrows can always
be rescaled so that $c_{Y,a}=1$ for all $Y$ and $a$.

For a $k$-linear category $C$ and a subset $\Sigma$ of its objects, the
\emph{object quotient} $C/[\Sigma]$ is obtained by quotienting by the ideal of
morphisms factoring through finite direct sums of objects in $\Sigma$ and
then removing those objects. In Sections~\ref{sec:scalar-normalization}
and~\ref{sec:mesh-deformations} we will restrict a scalar mesh category $C(Q,c)$ to a subset $S$ of vertices
that is closed under $\tau$ and $\tau^{-1}$ wherever defined (such as the
set of $\tau$-periodic vertices). Let $Q_S$ be the full translation
subquiver on $S$. Quotienting $C(Q,c)$ by the ideal generated by the
identity morphisms $1_x$ for $x\notin S$ and restricting to the objects in
$S$ gives a canonical isomorphism
\begin{equation}
 C(Q,c)/[Q_0\setminus S]
       \cong C(Q_S,c|_S).
 \label{eq:mesh-object-quotient}
\end{equation}
Indeed, the ideal of $kQ$ generated by the $1_x$ with $x\notin S$ is
spanned by precisely those paths that pass through a vertex outside $S$. Under this
quotient, a mesh relation $r_x$ with $x\notin S$ maps to zero, while a mesh
relation $r_x$ with $x\in S$ retains precisely the terms $(\sigma a)a$ whose
middle vertex lies in $S$, which is the mesh relation of $Q_S$ at $x$. In
particular, if $C(Q,c)$ is finite-dimensional, then so is $C(Q_S,c|_S)$.

\subsection{Exactness on radical layers}

We next show that the initial forms of the maps in an almost split
sequence induce exact sequences of graded Hom spaces in $\C$.
This exactness is the key to identifying
$\C$ with the mesh category $k(\Gamma_A)$ in
Proposition~\ref{prop:graded-realization}(3) and to reconstructing the module
category $\modu A_0$ in Proposition~\ref{prop:projective-evaluation}. Our
argument adapts Iyama's complementary-monomorphism technique for
$\tau$-categories \cite[Sections~3.3, 3.6 and~4.2]{IyamaI} directly to module
categories. The following criterion shows that a radical monomorphism
whose $\JJ^2$-perturbations remain monic raises radical degree by exactly
one under composition, providing injectivity at the left end of each graded
mesh sequence.

\begin{lemma}
\label{lem:strict-radical-layers}
Let $A$ be an Artin algebra, let $\JJ$ be the categorical radical of its
category of finite-length modules, and let $a:U\to Y$ be a radical
monomorphism. Suppose that $a+h$ is a monomorphism for every
$h\in \JJ^2(U,Y)$. Then for every finite-length module $W$, every
morphism $z:W\to U$, and every $n\geq1$,
\[
             a\circ z\in \JJ^n(W,Y)\quad\Longleftrightarrow\quad
             z\in \JJ^{n-1}(W,U).
\]
Furthermore, any finite direct sum of left almost split monomorphisms
satisfies the hypothesis that all such $\JJ^2$-perturbations remain
monomorphisms.
\end{lemma}
\begin{proof}
We first establish the perturbation property for almost split kernels, then
split off $a$ from a direct sum of right almost split maps and construct a
complementary monomorphism with the same property, and finally induct on the
radical degree.

\noindent\textbf{Step 1: Almost split kernels and the perturbation property.}
For any finite direct sum $Y$ of indecomposable modules, let $p:E_Y\to Y$ be
the direct sum of the minimal right almost split morphisms into its
indecomposable summands (using the radical inclusion $\rad P\hookrightarrow P$
at projective summands). By the lifting property of right almost split maps,
every radical morphism into $Y$ factors through $p$. For $n\geq0$, writing a
morphism in $\JJ^{n+1}(W,Y)$ as a sum of composites of $n+1$ radical morphisms
and factoring the last radical morphism through $p$ shows that
\begin{equation}
 p\circ \JJ^n(W,E_Y)=\JJ^{n+1}(W,Y)\qquad(n\geq0).
 \label{eq:right-radical-surjection}
\end{equation}
At each nonprojective indecomposable summand of $Y$, the kernel of $p$ is
the left almost split morphism of the corresponding almost split sequence,
while at each projective summand the inclusion $\rad P\hookrightarrow P$ has
zero kernel. Consequently, the kernel inclusion $i:T\to E_Y$ of $p$ is a
finite direct sum of left almost split monomorphisms.

More generally, let $i=\bigoplus_j i_j:T\to E$ be any finite direct sum of
left almost split monomorphisms $i_j:T_j\to E_j$. Any morphism in
$\JJ^2(T,Z)$ is a sum of composites $v\circ u$ with $u$ and $v$ radical. Since
the restriction of $u$ to each $T_j$ is radical, it factors through $i_j$ as
$u_j\circ i_j$, and $v\circ u_j$ remains radical. Thus
\[
 \JJ^2(T,Z)=\{q\circ i:q\in \JJ(E,Z)\}.
\]
Taking $Z=E$, any $h\in \JJ^2(T,E)$ has the form $h=q\circ i$ with
$q\in \JJ(E,E)$. Since $1+q$ is an automorphism of $E$, the perturbation
$i+h=(1+q)\circ i$ is again a monomorphism.

\noindent\textbf{Step 2: Splitting off the given monomorphism.}
Returning to the given monomorphism $a:U\to Y$, Step~1 shows that
$a$ factors through $p:E_Y\to Y$ as $a=p\circ u$ for some $u:U\to E_Y$. If
some component of $u$ between indecomposable summands is invertible, we may
place it in the upper left block and write
$u=\left(\begin{smallmatrix}u_{11}&u_{12}\\ u_{21}&u_{22}\end{smallmatrix}\right)$
with $u_{11}$ invertible. The elementary row and column operations
\[
 \begin{pmatrix}u_{11}^{-1}&0\\-u_{21}u_{11}^{-1}&1\end{pmatrix}
 \begin{pmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{pmatrix}
 \begin{pmatrix}1&-u_{11}^{-1}u_{12}\\0&1\end{pmatrix}
 =\begin{pmatrix}1&0\\0&u_{22}-u_{21}u_{11}^{-1}u_{12}\end{pmatrix}
\]
put that block in diagonal form, and repeating this elimination on the remaining
summands yields direct-sum decompositions $U=U''\oplus U'$ and
$E_Y=U''\oplus E'$ in which $u=1_{U''}\oplus u'$ and every component of $u'$
between indecomposable summands is noninvertible. After transporting $a$,
$p$, and $i$ along these coordinate automorphisms, we still have $a=p\circ u$
with $i$ the kernel inclusion of $p$, while the radical powers and the
$\JJ^2$-perturbation hypothesis are preserved. Since $u'\in\JJ(U',E')$ and
$p$ is radical, the restriction $a|_{U'}=p|_{E'}\circ u'$ lies in
$\JJ^2(U',Y)$. Setting $h=-a|_{U'}\circ\operatorname{pr}_{U'}\in \JJ^2(U,Y)$,
we find that $a+h$ vanishes on $U'$; since $a+h$ is a monomorphism by
hypothesis, $U'=0$ and $u$ is a split monomorphism. Choosing a complement
$V$ to $u(U)$ in $E_Y$, we identify $E_Y$ with $V\oplus U$ so that $u$ is the
inclusion of the second summand and write
\[
 E_Y=V\oplus U,\qquad p=(f,a),\qquad i=\binom b c,
\]
preserving both \eqref{eq:right-radical-surjection} and the kernel
factorization property of $i$.

\noindent\textbf{Step 3: The complementary monomorphism.}
We claim that the first component $b:T\to V$ of $i$ is also a radical
monomorphism satisfying the $\JJ^2$-perturbation hypothesis. First, $b$ and $c$
are radical because $i$ is radical. If $x\in\ker b$, then $i(x)=(0,c(x))$
and $0=p(i(x))=a(c(x))$; since $a$ is injective, $c(x)=0$ and hence
$i(x)=0$, so $x=0$. Next, given any $h\in \JJ^2(T,V)$, Step~1 lets us write
$h=q\circ i$ for a radical map $q=(q_V,q_U):V\oplus U\to V$. The upper
triangular matrix
\[
 L=\begin{pmatrix}1+q_V&q_U\\0&1\end{pmatrix}
\]
is an automorphism of $V\oplus U$, and the kernel inclusion of
$p\circ L^{-1}$ is $L\circ i=\binom{b+h}{c}$. On the other hand, the
restriction of $p\circ L^{-1}$ to $U$ is
$a-f\circ(1+q_V)^{-1}\circ q_U$, which differs from $a$ by a morphism in
$\JJ^2(U,Y)$ and is therefore a monomorphism by hypothesis. As just shown for
$p$ and $b$, the injectivity of the restriction of $p\circ L^{-1}$ to $U$
implies that the first component $b+h$ of its kernel inclusion is a
monomorphism.

\noindent\textbf{Step 4: Induction on the radical degree.}
We now prove by induction on $n\geq1$, simultaneously for all radical
monomorphisms $a$ satisfying the perturbation hypothesis, that
$a\circ z\in \JJ^n(W,Y)$ if and only if $z\in \JJ^{n-1}(W,U)$. Since $a$ is
radical, the ``if'' direction is immediate, as is the base case $n=1$. For
the induction step, suppose $a\circ z\in \JJ^{n+1}(W,Y)$ with $n\geq1$. By
\eqref{eq:right-radical-surjection}, there exist $v\in \JJ^n(W,V)$ and
$u\in \JJ^n(W,U)$ such that $a\circ z=p\circ\binom{v}{u}=f\circ v+a\circ u$.
Thus $\binom{v}{u-z}$ lies in the kernel of $p$, so
$\binom{v}{u-z}=\binom{b}{c}\circ s$ for some $s:W\to T$. In particular,
$b\circ s=v\in \JJ^n(W,V)$. Applying the induction hypothesis to the
complementary monomorphism $b$ yields $s\in \JJ^{n-1}(W,T)$. Since $c$ is
radical, it follows that $u-z=c\circ s\in \JJ^n(W,U)$, and hence
$z\in \JJ^n(W,U)$, completing the induction.
\end{proof}

Applying Lemma~\ref{lem:strict-radical-layers} and its dual to the maps in
each almost split sequence gives the following graded Hom-exactness
and mesh presentation. These are the finite-dimensional split forms of
Iyama's \cite[Theorems~5.2 and~9.2]{IyamaI} graded $\tau$-category and
mesh-species theorems: nilpotence of
the categorical radical makes completion unnecessary, and the
one-dimensional arrow spaces give scalar mesh relations. We give the
radical-layer argument in the present notation.

For a nonprojective object $Y$, let $\tau Y\xrightarrow{\bar\imath}E_Y
\xrightarrow{\bar p}Y$ denote the initial forms of the maps in its almost
split sequence. We call $\bar p$ the \emph{right mesh map} at $Y$.

\begin{proposition}[{\cite[Theorems~5.2 and~9.2]{IyamaI}}]
\label{prop:graded-realization}
Let $\C=\gr_\JJ\ind(\modu A)$. Then the following statements hold:
\begin{enumerate}[label={\upshape(\arabic*)}]
\item Let $0\to\tau Y\to E_Y\to Y\to0$ be an almost split sequence in
$\modu A$, and let $\tau Y\xrightarrow{\bar\imath}E_Y\xrightarrow{\bar p}Y$
be the sequence of initial forms of its morphisms in $\add\C$. For every object $W$ of $\C$, the
induced sequences
\[
 0\to\C(W,\tau Y)\to\C(W,E_Y)\to\C(W,Y),\qquad
 0\to\C(Y,W)\to\C(E_Y,W)\to\C(\tau Y,W)
\]
are exact as sequences of vector spaces. Both maps have degree one;
taking homogeneous components gives sequences with the corresponding
degree shifts. In particular, $\bar\imath$ is a monomorphism and $\bar p$ is an
epimorphism in $\add\C$.
\item For each indecomposable projective module $P$, the initial forms of
the components of the inclusion $\rad P\hookrightarrow P$ define a
monomorphism in $\add\C$. Dually, for each indecomposable injective module
$I$, the initial forms of the components of $I\twoheadrightarrow I/\soc I$
define an epimorphism in $\add\C$.
\item The chosen arrow representatives in $\C_1$ induce an isomorphism of
graded categories $k(\Gamma_A)\xrightarrow{\sim}\C$, where $k(\Gamma_A)$ has
the scalar mesh relations \eqref{eq:scalar-mesh}.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $i:\tau Y\to E_Y$ and $p:E_Y\to Y$ be the maps in the almost split
sequence ending at $Y$. By Lemma~\ref{lem:strict-radical-layers}, $i$ induces
an injection $\C_{n-1}(W,\tau Y)\hookrightarrow\C_n(W,E_Y)$ for every
$n\geq1$. Moreover, if $w\in \JJ^n(W,E_Y)$ ($n\geq1$) satisfies
$p\circ w\in \JJ^{n+2}(W,Y)$, then \eqref{eq:right-radical-surjection} yields
some $w'\in \JJ^{n+1}(W,E_Y)$ with $p\circ w'=p\circ w$. Since
$p\circ(w-w')=0$, we have $w-w'=i\circ z$ for some $z:W\to\tau Y$, and
$i\circ z\in \JJ^n(W,E_Y)$ implies $z\in \JJ^{n-1}(W,\tau Y)$ by
Lemma~\ref{lem:strict-radical-layers}. In degree $n=0$, any
$w\in\Hom_A(W,E_Y)$ with $p\circ w\in \JJ^2(W,Y)$ differs by an element of
$\JJ(W,E_Y)$ from a morphism in $\ker\Hom_A(W,p)$, which factors through the radical
map $i$ and therefore also lies in $\JJ(W,E_Y)$; thus $\bar p$ is injective on
$\C_0(W,E_Y)$. Since $p\circ i=0$, this proves exactness of the first
sequence in (1).

For an indecomposable projective module $P$, let $p:\rad P\hookrightarrow P$
be the inclusion. If $z\in \JJ^n(W,\rad P)$ ($n\geq0$) satisfies
$p\circ z\in \JJ^{n+2}(W,P)$, then \eqref{eq:right-radical-surjection} gives
$z'\in \JJ^{n+1}(W,\rad P)$ with $p\circ z'=p\circ z$, and injectivity of $p$
implies $z=z'\in \JJ^{n+1}(W,\rad P)$. Thus the initial form of $p$ is a
monomorphism in $\add\C$. Applying the dual arguments in the opposite module
category establishes exactness of the second sequence in (1) and the
statement for injective modules in (2).

Finally, sending each arrow of $\Gamma_A$ to its chosen class in $\C_1$
defines a graded functor $k\Gamma_A\to\C$. By \eqref{eq:scalar-mesh}, the
degree-two initial form of the relation $p\circ i=0$ is precisely the mesh
relation $r_Y$, so this functor descends to $k(\Gamma_A)\to\C$. It is full
because each radical power $\JJ^n$ is generated by products of $n$ morphisms
in $\JJ$. To see that it is faithful, we show by induction on $n\geq0$ that any
homogeneous combination $F$ of paths of length $n$ from $W$ to $Y$ whose
image vanishes in $\C_n(W,Y)$ belongs to the mesh ideal; the cases $n=0,1$
hold by definition of $\Gamma_A$. For $n\geq2$, we group $F$ in path order
by its final arrow
as $F=\sum_a F_a a$, where $a:X_a\to Y$. If $Y$ is nonprojective, the
components of $\bar p:E_Y\to Y$ are $v_a a$, so the tuple
$(F_a v_a^{-1})_a$ maps to zero under $\bar p:\C_{n-1}(W,E_Y)\to\C_n(W,Y)$.
By the exactness proved in (1), there is a path combination $H$ of length
$n-2$ from $W$ to $\tau Y$ whose image under $\bar\imath$ equals the image of
$(F_a v_a^{-1})_a$ in $\C_{n-1}(W,E_Y)$. Since the $a$-component of
$\bar\imath$ is $u_a(\sigma a)$ and $u_a v_a=c_{Y,a}$, each difference
$F_a - H c_{Y,a}(\sigma a)$ has length $n-1$ and maps to zero in $\C$. By
the induction hypothesis, each such difference lies in the mesh ideal, and
hence so does
\[
 F = H r_Y + \sum_a \bigl(F_a - H c_{Y,a}(\sigma a)\bigr)a.
\]
If $Y$ is projective, then $\bar p$ is a monomorphism by (2), so each $F_a$
maps to zero in $\C_{n-1}(W,X_a)$ and lies in the mesh ideal by induction.
\end{proof}

The cokernels of the right mesh maps are the simple functors on $\C$.
Together with Proposition~\ref{prop:graded-realization}(1), this gives
their minimal projective resolutions. We use these resolutions to check
the hypotheses of Lemma~\ref{lem:projective-evaluation-criterion} in the
proof of Proposition~\ref{prop:projective-evaluation}.

\begin{lemma}
\label{lem:mesh-simple-tops}
For every nonprojective object $X$ of $\C$, the image of the right mesh map
$\bar p:\C(-,E_X)\to\C(-,X)$ is the radical $\bigoplus_{n\geq1}\C_n(-,X)$ of
the representable functor $\C(-,X)$, and its cokernel is the simple functor
$\C_0(-,X)$. The same conclusion holds at each projective object $P$ for the
map $\C(-,\rad P)\to\C(-,P)$ of Proposition~\ref{prop:graded-realization}(2).
\end{lemma}
\begin{proof}
Let $p:E_X\to X$ be the minimal right almost split map in $\modu A$ (or the
inclusion $\rad X\hookrightarrow X$ when $X$ is projective). By
\eqref{eq:right-radical-surjection}, $p\circ \JJ^n(W,E_X)=\JJ^{n+1}(W,X)$ for
every $n\geq0$ and every object $W$. Passing to the associated graded spaces
shows that the image of $\bar p$ on $\C(W,E_X)$ is precisely the
positive-degree ideal $\bigoplus_{n\geq1}\C_n(W,X)$. Since this ideal is
nilpotent and the degree-zero quotient has $\C_0(X,X)=k$ and $\C_0(W,X)=0$
for $W\ne X$, the positive-degree part is the radical of $\C(-,X)$ and its
cokernel is the simple functor $\C_0(-,X)$.
\end{proof}

\subsection{Evaluation on projective objects}

Following Bretscher and Gabriel \cite[Section~3.1]{BretscherGabriel}, we now
define the standard form $A_0$ as the opposite endomorphism algebra of the
projective objects in $\C$, and we show using the graded exactness properties
above that evaluating objects of $\add\C$ on the projectives gives an
equivalence $\add\C\simeq\modu A_0$.

\begin{construction}
\label{con:projective-algebra}
Let $\D$ be the full subcategory of $\C$ on the objects $P_x=e_x A$
corresponding to the indecomposable projective $A$-modules. We define the
graded algebra $A_0=\bigoplus_{x,y}A_0(x,y)$ by
\[
 A_0(x,y)=\D(P_y,P_x)
       =\bigoplus_{n\geq0}\JJ^n(P_y,P_x)/\JJ^{n+1}(P_y,P_x).
\]
For $b\in A_0(x,y)$, denote the corresponding morphism in $\D(P_y,P_x)$
by $\operatorname{lm}_b$. Define multiplication by
\[
 \operatorname{lm}_{bc}=\operatorname{lm}_b\circ\operatorname{lm}_c
 \qquad(b\in A_0(x,y),\ c\in A_0(y,z)).
\]
Thus $A_0=\End_{\C}(\bigoplus_x P_x)^{\op}$ under our path-order convention
for endomorphism algebras. Under $A(x,y)\cong\Hom_A(P_y,P_x)$,
the filtration $F^nA(x,y)=\JJ^n(P_y,P_x)$ has associated graded algebra
$\gr_F A=A_0$, with the multiplication just defined.
The \emph{projective evaluation
functor} sends each object $X\in\add\C$ to the right $A_0$-module
$\bigoplus_x\C(P_x,X)$, on which $b\in A_0(x,y)$ acts on $m\in\C(P_x,X)$ by
$mb=m\circ\operatorname{lm}_b\in\C(P_y,X)$.
\end{construction}

To prove that projective evaluation is an equivalence of categories, we use a
standard homological realization criterion from the Auslander correspondence
(compare \cite[Theorem~3.3]{IyamaRealization}).

\begin{lemma}
\label{lem:projective-evaluation-criterion}
Let $\Lambda$ be a finite-dimensional algebra over a field, let $e\in\Lambda$
be an idempotent, and set $\Lambda_e=e\Lambda e$. Suppose that
\[
 \Hom_\Lambda(T,\Lambda)=\Ext^1_\Lambda(T,\Lambda)=0
 \qquad\text{for every simple $\Lambda$-module $T$ with }Te=0.
\]
Then the functor $P\mapsto Pe$ is fully faithful on $\proj\Lambda$. If, in
addition, $\operatorname{gl.dim}\Lambda\leq2$ and every finite-dimensional
injective $\Lambda_e$-module is isomorphic to $Qe$ for some
$Q\in\proj\Lambda$, then $P\mapsto Pe$ is an equivalence of categories
$\proj\Lambda\xrightarrow{\sim}\modu\Lambda_e$, and in particular
$\Lambda_e$ is representation-finite.
\end{lemma}
\begin{proof}
Induction on composition length extends the assumed vanishings to every
finite-dimensional module $T$ with $Te=0$. For $M\in\modu\Lambda$,
the kernel and cokernel of
$(Me)\otimes_{\Lambda_e}e\Lambda\to M$ are annihilated by $e$. Hence, for
$Q\in\proj\Lambda$, these vanishings and tensor--Hom adjunction give
\[
 \Hom_\Lambda(M,Q)\xrightarrow{\sim}
 \Hom_\Lambda((Me)\otimes_{\Lambda_e}e\Lambda,Q)
 \xrightarrow{\sim}\Hom_{\Lambda_e}(Me,Qe)
\]
by restriction to $Me$. In particular, $P\mapsto Pe$ is fully faithful
on $\proj\Lambda$.

Under the additional hypotheses, take an injective copresentation
$0\to X\to I^0\to I^1$ over $\Lambda_e$ and lift $I^0\to I^1$ to
$f:Q^0\to Q^1$ in $\proj\Lambda$. The second syzygy $\ker f$ is
projective because $\operatorname{gl.dim}\Lambda\leq2$, and exactness of
$(-)e$ gives $X\cong(\ker f)e$. This proves essential surjectivity.
Representation-finiteness follows from the finite number of
indecomposable projective $\Lambda$-modules.
\end{proof}

We now verify the hypotheses of Lemma~\ref{lem:projective-evaluation-criterion}
for $\add\C$ and $A_0$.

\begin{proposition}
\label{prop:projective-evaluation}
Let $A$ be a basic representation-finite $k$-algebra. Let $\C$ and $A_0$
be the graded category and algebra of
Construction~\ref{con:projective-algebra}.
\begin{enumerate}[label={\upshape(\arabic*)}]
\item The projective evaluation functor induces an equivalence of categories
$\add\C\xrightarrow{\sim}\modu A_0$.
\item The algebra $A_0$ is basic and representation-finite.
\item If $X_0\in\modu A_0$ denotes the evaluation of an object $X\in\C$, then
\[
                         \dim_k X_0e_x=\dim_k Xe_x
\]
for every primitive idempotent $e_x$.
\end{enumerate}
\end{proposition}
\begin{proof}
Because the radical filtration on $\Hom_A(X,Y)$ is finite, passing to the
associated graded vector space preserves total dimension:
\begin{equation}
 \dim_k\C(X,Y)=\sum_{n\geq0}\dim_k \JJ^n(X,Y)/\JJ^{n+1}(X,Y)
             =\dim_k\Hom_A(X,Y).
 \label{eq:graded-hom-dimensions}
\end{equation}
In particular, $\dim_k X_0 e_x=\dim_k\C(P_x,X)=\dim_k\Hom_A(P_x,X)=\dim_k Xe_x$,
which proves (3).

For (1) and (2), set $G=\bigoplus_{X\in\operatorname{ob}\C}X$ and
$\Lambda=\End_{\add\C}(G)^{\op}$. The Yoneda embedding identifies $\add\C$
with $\proj\Lambda$ via $R_X=\C(G,X)$, sending the simple tops to
$L_X=\C_0(G,X)$. Letting $e\in\Lambda$ be the idempotent projecting onto the
direct sum of the projective objects $P_x$, we have $e\Lambda e=A_0$ and
$\operatorname{ev}(M)=Me$, so we may apply
Lemma~\ref{lem:projective-evaluation-criterion}.

For each nonprojective object $X$, Proposition~\ref{prop:graded-realization}(1)
and Lemma~\ref{lem:mesh-simple-tops} yield the projective resolution
\[
 0\longrightarrow R_{\tau X}\longrightarrow
 \bigoplus_{Y\to X}R_Y\longrightarrow R_X\longrightarrow L_X
 \longrightarrow0,
\]
while for $P_x$ they yield $0\to R_{\rad P_x}\to R_{P_x}\to L_{P_x}\to0$;
thus $\operatorname{gl.dim}\Lambda\leq2$. Applying $\Hom_\Lambda(-,\Lambda)$
to the resolution of $L_X$ for nonprojective $X$ produces the sequence
$0\to\C(X,G)\to\C(E_X,G)\to\C(\tau X,G)$, which is exact by
Proposition~\ref{prop:graded-realization}(1). Consequently,
$\Hom_\Lambda(L_X,\Lambda)=\Ext^1_\Lambda(L_X,\Lambda)=0$ for every simple
module $L_X$ annihilated by $e$, and
Lemma~\ref{lem:projective-evaluation-criterion} shows that $\operatorname{ev}$
is fully faithful on $\proj\Lambda$.

Let $S_x=P_x/\rad P_x$ and $I_x=D_k(Ae_x)$ denote the simple and
indecomposable injective $A$-modules at vertex $x$. By
\eqref{eq:graded-hom-dimensions}, $S_x^0=\operatorname{ev}(R_{S_x})$
is the simple $A_0$-module at $x$. Set $N_x=\operatorname{ev}(R_{I_x})$.
Full faithfulness gives
\[
 \dim_k\Hom_{A_0}(S_y^0,N_x)=\dim_k\C(S_y,I_x)=\dim_k\Hom_A(S_y,I_x)=\delta_{xy},
\]
which shows that $\soc N_x\cong S_x^0$. On the other hand,
\eqref{eq:graded-hom-dimensions} gives
\[
 \begin{aligned}
 \dim_k N_xe_y&=\dim_k\C(P_y,I_x)=\dim_k I_xe_y=\dim_k e_yAe_x\\
 &=\dim_k\C(P_x,P_y)=\dim_k e_yA_0e_x=\dim_k D_k(A_0 e_x)e_y.
 \end{aligned}
\]
Since $N_x$ has simple socle $S_x^0$, it embeds into the injective hull
$D_k(A_0 e_x)$, and equality of dimensions implies $N_x\cong D_k(A_0 e_x)$.
Thus every indecomposable injective $A_0$-module lies in the image of
$\operatorname{ev}$. Lemma~\ref{lem:projective-evaluation-criterion} now
implies that $\add\C\xrightarrow{\sim}\modu A_0$ is an equivalence. Since
the objects $P_x$ are pairwise nonisomorphic in $\C$, the algebra $A_0$ is
basic as well as representation-finite.
\end{proof}

The equivalence also preserves the almost split sequences. We will use
this identification for the graded extension bounds in
Lemma~\ref{lem:positive-ext-vanishing}.

\begin{corollary}
\label{cor:same-ar-quiver}
The Auslander--Reiten quivers of $A_0$ and $A$ are isomorphic as translation
quivers: $\Gamma_{A_0}\cong\Gamma_A$.
\end{corollary}
\begin{proof}
By Proposition~\ref{prop:projective-evaluation}(1), we may compute
$\Gamma_{A_0}$ directly inside $\add\C$. Since $\C$ is generated in degree
one, the radical of $\add\C$ is its positive-degree ideal and the radical
square is the ideal of degree at least two; hence the arrows of $\Gamma_A$
form bases of the spaces of irreducible morphisms in $\add\C$. By
Lemma~\ref{lem:mesh-simple-tops}, each right mesh map $\bar p:E_X\to X$ is
right almost split in $\add\C$. Because the indecomposable summands of $E_X$
are pairwise nonisomorphic by Lemma~\ref{lem:square-free}, every endomorphism
$\varphi$ of $E_X$ is diagonal modulo the radical. If
$\bar p\circ\varphi=\bar p$, linear independence of the arrow components
modulo the radical square implies that all diagonal entries of $\varphi$
are $1$. Thus every such endomorphism is an
automorphism, which means that $\bar p$ is right minimal. The mesh sequences
in $\add\C$ are therefore the almost split sequences of $\modu A_0$, so the
translation of $\Gamma_{A_0}$ agrees with that of $\Gamma_A$.
\end{proof}